Problem 13 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 7-x \geq 5 $$

Step-by-Step Solution

Answer

The solution is \( x \leq 2 \), in interval notation \((- fty, 2]\).

Step 1: Isolate x

First, we need to isolate the variable \( x \) on one side of the inequality. Starting with the inequality \( 7 - x \geq 5 \), subtract 7 from both sides to get it closer to isolating \( x \). This will give us: \( -x \geq 5 - 7 \). Simplifying this, we have \( -x \geq -2 \).

Step 2: Solve for x

Next, solve for \( x \) by dividing both sides by \(-1\). Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. Therefore, dividing \(-x \geq -2 \) by \(-1\) gives us \( x \leq 2 \).

Step 3: Express in Interval Notation

Now, express the solution \( x \leq 2 \) in interval notation. Since \( x \) can be any number less than or equal to 2, the interval is \((-fty, 2]\).

Step 4: Graph the Solution

To graph \( x \leq 2 \), draw a number line. Place a closed dot on 2 to indicate that it is included in the solution set. Shade the number line to the left of 2 to indicate all numbers less than 2 are included.

Key Concepts

Solving Inequalities

Inequalities are similar to equations, but instead of using the equal sign, they use inequality symbols like \( >, <, \geq, \leq \). Solving inequalities involves finding the set of values that make the inequality true. The process generally follows the same principles as solving equations, with a few key differences.

• First, we need to isolate the variable on one side of the inequality.
• Perform arithmetic operations such as addition or subtraction to both sides, ensuring the inequality remains true.
• If you multiply or divide both sides by a negative number, remember to reverse the inequality sign.

For example, in the inequality \( 7-x \geq 5 \), we initially isolate \( x \) by subtracting 7 from both sides, resulting in \( -x \geq -2 \). Then, we solve for \( x \) by dividing each side by \(-1\), reversing the inequality to get \( x \leq 2 \). This tells us \( x \) can be any number less than or equal to 2.

Inequality Graphing

Graphing an inequality helps to visually represent the solution set on a number line, making it easier to see which values \( x \) can take. Here's how you can graph inequalities:

• Draw a straight horizontal line which serves as the number line.
• Identify the critical number(s); in our example, it's 2.
• If the inequality is strict (\(<\) or \(>\)), place an open circle on the number to show it's not included. If it allows equality (\(\leq\) or \(\geq\)), use a closed dot to show inclusion.
• Shade the section of the number line representing all potential solution values.

For \( x \leq 2 \), place a closed dot at 2, since 2 is part of the solution, and shade everything to the left, indicating values less than 2 are also valid solutions.

Interval Notation

Once you have the solution set for an inequality, it's common to express this set using interval notation. This notation is a concise way to describe ranges of numbers:

• Use parentheses \( () \) for solutions that do not include the endpoint (non-inclusive).
• Use brackets \([] \) for solutions that do include the endpoint (inclusive).
• The infinity symbol \((\infty)\) is always paired with parentheses, as infinity itself is not a number and cannot be an endpoint.

For instance, the solution \( x \leq 2 \) translates to the interval \((-\infty, 2]\). This indicates that \( x \) can be any value up to and including 2, extending indefinitely to negative infinity.